## Experimental Mathematics

### Congruence Subgroups of Groups Commensurable with PSL$(2,\Z)$ of Genus 0 and 1

C. J. Cummins

#### Abstract

Thompson has shown that up to conjugation there are only finitely many congruence subgroups of $\PSL(2,\R)$ of fixed genus. For $\PSL(2,\Z)$, Cox and Parry found an explicit bound for the level of a congruence subgroup in terms of its genus. This result was used by the author and Pauli to compute the congruence subgroups of $\PSL(2,\Z)$ of genus less than or equal to 24. However, the bound of Cox and Parry applies only to $\PSL(2,\Z)$. In this paper a result of Zograf is used to find a bound for the level of any congruence subgroup in terms of its genus. Using this result, a list of all congruence subgroups, up to conjugacy, of $\PSL(2,\R)$ of genus 0 and 1 is found.

This tabulation is used to answer a question of Conway and Norton who asked for a complete list of genus 0 subgroups, $\overline G$, of $\PSL(2,\R)$ such that

(i) $\overline G$ contains $\overline\Gamma_0(N)$ for some $N$.

(ii) $\overline G$ contains the translation $z\mapsto z+k$ iff $k$ is an integer.

Thompson has also shown that for fixed genus there are only finitely many subgroups of $\PSL(2,\R)$ which satisfy these conditions. We call these groups "moonshine groups.'' The list of genus 1 moonshine groups is also found. All computations were performed using Magma.

#### Article information

Source
Experiment. Math., Volume 13, Issue 3 (2004), 361-382.

Dates
First available in Project Euclid: 22 December 2004

https://projecteuclid.org/euclid.em/1103749843

Mathematical Reviews number (MathSciNet)
MR2103333

Zentralblatt MATH identifier
1099.11022

#### Citation

Cummins, C. J. Congruence Subgroups of Groups Commensurable with PSL$(2,\Z)$ of Genus 0 and 1. Experiment. Math. 13 (2004), no. 3, 361--382. https://projecteuclid.org/euclid.em/1103749843